We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph G is a set S of vertices such that each vertex is
either in S or has a neighbour in S. In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions s and t such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of k, we consider properties of Dk(G), the graph
consisting of a vertex for each dominating set of size at most k and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that DΓ(G)+1(G) is not necessarily
connected, for Γ(G) the maximum cardinality of a minimal dominating set
in G. The result holds even when graphs are constrained to be planar, of
bounded tree-width, or b-partite for b≥3. Moreover, we construct an
infinite family of graphs such that Dγ(G)+1(G) has exponential
diameter, for γ(G) the minimum size of a dominating set. On the positive
side, we show that Dn−m(G) is connected and of linear diameter for any
graph G on n vertices having at least m+1 independent edges.Comment: 12 pages, 4 figure